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A211392 The number of divisors d of n! such that the symmetric group on n letters contains no elements of order d.

%I A211392 #25 Nov 22 2023 21:38:42
%S A211392 0,0,1,4,10,24,51,85,146,254,520,769,1557,2561,3997,5333,10705,14633,
%T A211392 29315,40970,60722,95912,191902,242769,339909,532088,677224,917112,
%U A211392 1834373,2332596,4665375,5529352,7864049,12164824,16422587,19595164,39190653,60465758
%N A211392 The number of divisors d of n! such that the symmetric group on n letters contains no elements of order d.
%H A211392 Alois P. Heinz, <a href="/A211392/b211392.txt">Table of n, a(n) for n = 1..1000</a>
%F A211392 a(n) = A000005(n!) - A009490(n).
%p A211392 b:= proc(n,i) option remember; local p;
%p A211392       p:= `if`(i<1, 1, ithprime(i));
%p A211392       `if`(n=0 or i<1, 1, b(n, i-1)+
%p A211392       add(b(n-p^j, i-1), j=1..ilog[p](n)))
%p A211392     end:
%p A211392 a:= n-> numtheory[tau](n!) -b(n, numtheory[pi](n)):
%p A211392 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 15 2013
%t A211392 b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n==0 || i<1, 1, b[n, i-1] + Sum[b[n-p^j, i-1], {j, 1, Floor@Log[p, n]}]]];
%t A211392 a[n_] := DivisorSigma[0, n!] - b[n, PrimePi[n]];
%t A211392 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Mar 24 2017, after _Alois P. Heinz_ *)
%Y A211392 Cf. A000005, A009490, A027423, A211391.
%K A211392 nonn
%O A211392 1,4
%A A211392 _Alexander Gruber_, Feb 07 2013
%E A211392 More terms from _Alois P. Heinz_, Feb 11 2013